Prices, Profits, and Preference Dependence∗

Prices, Pro…ts, and Preference Dependence
Yongmin Cheny and Michael H. Riordanz
Revised December 2014
Abstract: We develop a new approach to discrete choice demand for di¤erentiated
products, using copulas to separate the marginal distribution of consumer values
for product varieties from their dependence relationship, and apply it to the issue
of how preference dependence a¤ects market outcomes in symmetric multiproduct
industries. We show that greater dependence lowers prices and pro…ts under certain
conditions, suggesting that preference dependence is a distinct indicator of product
di¤erentiation. We also …nd new su¢ cient conditions for the symmetric multiproduct
monopoly and the symmetric single-product oligopoly prices to be above or below the
single-product monopoly price.
Keywords: Product di¤erentiation, discrete choice, copula, multiproduct industries.
Earlier versions, circulated under the titles "Preference and Equilibrium in Monopoly
and Duopoly" and "Preferences, Prices, and Performance in Multiproduct Industries", were presented at the 2009 Summer Workshop in Industrial Organization (University of Auckland), 2009 Summer Workshop on Antitrust Economics and Competition Policy (SHUFE, Shanghai), 2009 IO Day Conference (NYU), Segundo Taller de
Organización Industrial (Chile, 2009), 2010 Choice Symposium (Key Largo), and seminars at Ecole Polytechnique/CREST, Fudan University, Lingnan University, Mannheim
University, National University of Singapore, University of Melbourne, and University of Rochester. The authors thank conference and seminar participants, especially
discussants Simon Anderson and Barry Nalebu¤, and several referees for useful comments.
y
University of Colorado at Boulder; [email protected]
zColumbia University; [email protected]
I. INTRODUCTION
A key issue in the economics of di¤erentiated product markets is how the relationship between consumer values for di¤erent product varieties matters for market
outcomes. Typical discrete choice models of product di¤erentiation assume either
that consumer values for di¤erent product varieties are independent (e.g. Perlo¤ and
Salop, 1985) or follow a joint distribution of a speci…c form (e.g., Anderson, et al.,
1992). Such special cases are insightful but lack the general structure needed for a
more complete understanding of how the correlation of consumer values for alternative
products a¤ects market outcomes.
In this paper, we develop a new approach to discrete choice demand in di¤erentiated product markets that is more general than the familiar approaches. The key
feature of the new approach is to use copulas to separate the marginal distribution
of consumer values for each variety from their dependence relationship. A copula is
a multivariate uniform distribution that “couples” marginal distributions to form a
joint distribution. Furthermore, by Sklar’s Theorem, it is without loss of generality
to represent a joint distribution of consumer values by its marginal distributions and
a copula (Nelsen, 2006). The virtue of the copula approach is that all information
about dependence (or correlation) of values is contained in the copula. Thus the copula representation of consumer preferences makes it straightforward to analyze how
market outcomes are a¤ected by the distribution of values for each variety holding the
dependence relationship constant, or by the dependence properties among the values
for arbitrary marginal distributions. In this way, the copula approach provides an
elegant and useful representation of consumer preferences for di¤erentiated products.
In Section 2, we present a model of consumer preferences over an arbitrary number
of symmetric varieties of a good. Consumer values for the varieties are assumed
to follow a smooth and symmetric joint probability distribution. We interpret the
mean and variance of the marginal distribution as measures of preference strength
and preference diversity respectively, and let the preference dependence properties of
the copula capture the correlations of values. We de…ne preference dependence using
standard concepts of positive and negative dependence of random variables, and order
copulas accordingly. We apply this approach to investigate two issues.
First, in Section 3, we study how prices and pro…ts change with the degree of pref2
erence dependence in symmetric multiproduct markets. In discrete choice models
of product di¤erentiation that assume independence between values of di¤erent varieties, a higher variance of consumer values (i.e. preference diversity) raises price and
pro…t under certain conditions, and thus can be interpreted as an indicator of product
di¤erentiation (e.g., Anderson et. al. al, 1992; Johnson and Myatt, 2006; Perlo¤ and
Salop, 1985). A natural question to ask is, when preferences for di¤erent varieties are
not independent, how does preference dependence relate to product di¤erentiation?
Intuitively, greater dependence means that more consumers regard the varieties as
closer substitutes, suggesting that product di¤erentiation is less when preference dependence is greater. We …nd that price and pro…t decrease in preference dependence
for a symmetric multiproduct monopoly or a symmetric single-product oligopoly under certain conditions. Therefore, preference depedenence can be interpreted as a
distinct indicator of product di¤erentiation, separate from preference diversity.
Second, in Section 4, we examine how prices di¤er across several market structures. This issue is relevant in various scenarios: (1) as a result of innovation, a
single-product monopolist introduces new varieties to become a multiproduct monopolist; (2) as the result of lower entry barriers, such as expiration of a patent,
competing single-product …rms enter a previously monopolized market; (3) due to a
change in merger policy, a single-product symmetric oligopoly merges into a multiproduct monopoly. Understanding the price e¤ects in such scenarios is both theoretcally interesting and policy relevant. While Chen and Riordan (2008) analyzes the
issue for the special case in which the marginal distribution of consumer values is
exponential, the present paper provides general comparisons for arbitrary marginal
distributions. Speci…cally, we …nd that the single-product monopoly price is higher
than the symmetric oligopoly price if the hazard rate of the marginal distribution
is non-decreasing and preferences are positively dependent, but lower if the hazard
rate is non-increasing and preferences are negatively dependent.1 Furthermore, the
symmetric multiproduct monopoly price is higher than the single-product monopoly
price when preferences are uniformly positively dependent or negatively dependent.
1
As we can see from this result, the copula approach is more powerful than simply assuming
a particular joint distribution of consumer values for alternative products. The bivariate normal
distribution, for example, does have the virtue of neatly separating preference diversity (variance)
and preference dependence (correlation), but is restrictive in part because the marginal normal
distribution has a particular shape with an increasing hazard rate.
3
We make concluding remarks in Section 5, and gather proofs in the Appendix.
II. THE COPULA REPRESENTATION OF PREFERENCES
Consumers are assumed to purchase at most one of n
2 possible varieties of
a good. A consumer’s value (or willingness to pay) for the ith variety is wi . To
describe consumer preferences, the standard approach is to specify a joint distribution
of w
(w1 ; :::; wn ). The analyses typically proceed by assuming independence or
that the joint distribution has a particular function form, e.g. is a bivariate normal
distribution. For tractability, it is often also assumed that the joint distribution
function is symmetric. The copula approach to discrete choice demand provides a
more general structure for modeling the correlation of consumer values for di¤erent
varieties, while remaining tractable in the symmetric case.
The copula representation of preferences is based on Sklar’s Theorem in statistics,
which states that the probability distribution of a vector of random variables can be
represented by a copula and marginal distributions. More speci…cally, a copula is
a multivariate cumulative distribution function with uniform marginal distribution
functions.2 According to Sklar’s Theorem, if H (w) is a multivariate distribution
function with marginal distribution functions Fi (wi ), then there exists a copula C(x)
such that H (w) = C (F1 (w1 ) ; :::; Fn (wn )). Conversely, if Fi (wi ) for i = 1; :::; n
are univariate distribution functions and C (x) is a copula, then the composition
H (w) = C (F1 (w1 ) ; :::; Fn (wn )) is a multivariate distribution function with marginal
distribution functions Fi (wi ).
Consider a population of consumers whose size is normalized to 1, and assume for
simplicity that the joint distribution of w in the population is symmetric, and also
that the marginal distribution function for each variety can be inverted to obtain a
strictly-increasing continuous function wi = w (xi ). Then, by construction, xi is distributed uniformly on the unit interval I [0; 1]. Conversely, by Sklar’s Theorem, the
2
A copula C (x) is an n-increasing function, de…ned for all x
C(x; 1; :::1) = :::: = C (1; :::1; x) = x
and
C(x1 ; :::xn
1 ; 0)
= ::: = C(0; x2 ; :::xn ) = 0:
See Nelsen (2006).
4
n
(x1 ; :::; xn ) 2 [0; 1] ; satisfying
symmetric joint distribution of consumer values for the n varieties is fully described
by a continuous strictly-increasing valuation function w (xi ) and a symmetric copula
C (x). A consumer’s type can be thought of as a point in the unit cube, x 2 I n , with
the copula describing the population of types. Finally, assume for further simplicity
that w (xi ) and C (x) are twice di¤erentiable functions.
The copula approach to representing consumer preferences models the strength and
dispersion of consumer tastes for individual varieties separately from the correlation
of tastes for di¤erent varieties. Under the simplifying symmetry, monotonicity, and
smoothness assumptions, the inverse of the valuation function de…nes the marginal
distributions of consumer values, while the copula contains all of the information
about the correlation of values. The approach enables a general treatment of how
correlation (or dependence) matters, based on the properties of the symmetric copula,
while still maintaining a tractable analysis, and without unduly restricting the smooth
symmetric distribution functions under consideration.
The advantage of the copula representation of consumer preferences over the standard approach is to separate cleanly the dependence properties of the joint distribution of values from the properties of the marginal distributions. This enables
us to investigate how the correlation of consumer values matters for market outcomes for a wider class of joint distributions, since di¤erent marginal distributions
generate di¤erent joint distributions for a given copula. As noted above, the copula determines the statistical dependence of consumer values for the varieties. In
particular, C (x) = i=1;:::;n xi is the independence copula, and C (x) is positively
(negatively) orthant dependent if C (x) > (<) i=1;:::;n xi for all x 2 (0; 1)n . Furthermore, C1 (x) @C (x) =@x1 is the conditional distribution of (x2 ; :::; xn ) given x1 , and
C11 (x)
@ 2 C (x) =@x21 < (>) 0 for all x 2 (0; 1)n indicates positive (negative) stochastic dependence.3 Because marginal distribution functions are monotonic, these
properties of the copula translate directly into corresponding dependence properties
of the joint distribution of consumer values. For example, positive orthant dependence means that the probability that a randomly drawn consumer’s values for all
varieties are high (or low) is greater than if the values were independent, and positive stochastic dependence means that a high realization of w1 shifts the conditional
3
By symmetry, these stochastic dependence properties can also be de…ned using Ck ( ) and Ckk ( )
for k = 2; :::; n:
5
distribution of (w2 ; :::; wn ) according to …rst-order stochastic dominance. In what follows, we shall say simply that consumer values are positively dependent or negatively
dependent when both the appropriate orthant and stochastic dependence conditions
are satis…ed.
It is convenient for our purposes to consider an arbitrary family of copulas indexed
by a parameter . The copula family is ordered by increasing orthant dependence if
a higher indicates greater orthant dependence, i.e. C (x) @C (x; ) =@ > 0 for
interior x. Similarly, the copula family is ordered by increasing stochastic dependence
if C11 (x; ) @C11 (x; ) =@ < 0 for interior x. Roughly speaking, greater orthant
dependence means that there is a lower probability that consumers have low values
for some products and high values for the others, while greater stochastic dependence
means that a higher value for one variety makes low values for the others more likely.
We will refer to the orthant dependence and stochastic orders collectively as increasing
dependence.4
We use the properties of the valuation function and the copula to measure consumer preferences along three dimensions: preference strength, preference diversity,
and preference dependence. Preference strength refers to how much consumers on average value each variety, while preference diversity refers to the heterogeneity of those
values. The mean and variance of consumer values for each variety are, respectively,
R1
R1
2
w
(x)
dx
and
[w (x)
]2 dx: We interpret to measure preference
0
0
strength and to measure preference diversity. Both are properties of the marginal
distribution, and has been considered an indicator of the degree of product di¤erentiation under the assumption that consumer values are independent (Perlo¤ and
Salop, 1985). Preference dependence refers to the correlation of consumer values for
di¤erent varieties, and is measured by a parameter , indexing an ordered family of
copulas. A higher value of indicates that the values for di¤erent varieties are more
positively dependent or less negatively dependent. We argue below that can be
interpreted as a distinct indicator of product di¤erentiation.
Given the copula representation of preferences, it is straightforward to derive consumer demand. Denote the price for good 1 by p; and the prices for the rest of the
4
If n = 2; then C11 (x) < 0 implies C (x) > 0 (Nelsen, 2006). Nelsen (2006) discusses various
copula families and their dependence properties for the case of n = 2; see also Joe (1997) for related
discussions.
6
n 1 goods by ri : It is convenient to normalize the consumer values and the prices
by de…ning
ui
wi
;
;
p
p
;
ri
ri
:
Moreover, denoting the marginal distribution of ui = w(xi ) by F (ui ), by the Sklar’s
Theorem, the joint distribution of normalized values is C (F (u1 ) ; :::; F (un )). A type
x consumer will purchase good 1 under the following condtions:
x1
F (u (x1 )
Therefore, de…ning u (xi )
Q (p; r2 ; :::; rn ) =
Z
F
1
F (p) ;
p + ri )
xi ; i = 2; :::; n:
(xi ), the demand for good 1 is
1
C1 (x1 ; F (u (x)
p + r2 ) ; :::; F (u (x1 )
p + rn )) dx1 :
(1)
F (p)
The demand for other goods is derived similarly. It follows that any two goods are
always substitutes because, for j = 2; :::; n;
@Q (p; r2 ; :::; rn )
@rj
1
Z
=
C1j (x; F (u (x)
p + r2 ) ; :::; F (u (x)
p + rj )) f (u (x)
p + rj ) dx > 0;
F (p)
where f (ui ) is the density function. If only a single good is o¤ered, then its demand
is simply Q(p) = 1 F (p).
We conclude this section by introducing the Farlie-Gumble-Morgenstern (FGM)
copula family. Its general form for n 3 is given in Nelsen (2006). When n = 2, it
becomes
C (x1 ; x2 ) = x1 x2 + x1 x2 (1 x1 )(1 x2 ):
Our results in the next two sections can be illustrated with examples that combine an
FGM copula for n = 2 with the exponential marginal distribution F (ui ) = 1 e ui 1 :
7
We shall refer to this as the FGM-exponential case. More details of these illustrative
examples are contained in Chen and Riordan (2011).
III. IMPACT OF DEPENDENCE ON PRICE AND PROFIT
In this section, we consider how preference dependence a¤ects price and pro…t.
We maintain three additional simplifying assumptions for this and the next section.
First, the average cost of production for each variety is constant, and without loss of
generality normalized to zero. An appropriate interpretation of the normalization is
that consumers reimburse the …rm for the cost of producing the product in addition
to paying a markup p. Consequently, can be interpreted as mean value minus
constant average variable cost, and thus can be either positive or negative. Second,
at least some consumers have positive values so that there are gains from trade, i.e.
w (1) > 0. Third, equilibrium prices exist uniquely and are interior under all market
structures, and they are symmetric under multiproduct monopoly or oligopoly when
n 2.5
As a benchmark, we …rst note that the single-product monopoly (gross) pro…t
function is m (p) = (p + ) [1 F (p)] : The pro…t-maximizing normalized price
(pm ) satis…es the …rst-order condition
(pm + ) (pm ) = 1
(2)
and the second-order condition
(pm + )
0
(pm ) + (pm ) > 0
f (u)
is the hazard rate determining the
at an interior solution, where (u)
1 F (u)
elasticity of demand. A standard regularity condition, for which an increasing hazard
rate ( 0 (u) 0) is su¢ cient but not necessary, guarantees a unique interior maximum:
d [(u + ) (u)] =du > 0.
5
For convenience, we refer to optimal prices under monopoly as equilibrium prices. An interior
price satis…es p 2 (w(0); w(1)), so the market is neither shut down nor fully covered. Consequently,
pro…t functions are di¤erentiable at equilibrium prices. Given the symmetry of C ( ) ; the symmetric
price assumption is quite natural; it is satis…ed, for example, in our FGM-uniform case.
8
Next, we consider a price-setting multiproduct monopoly producing n
metric varieties of the good. Its pro…t function for a symmetric price is
mm
(p) = (p + ) [1
C (F (p); :::F (p); )] :
2 sym-
(3)
The pro…t-maximizing normalized price pmm satis…es
(pmm + )
and
(pmm + )
where
C
(u; )
d
C
C
(pmm ; ) = 1
(pmm ; )
+
du
C
(4)
(pmm ; ) > 0
(5)
nC1 (F (u) ; :::; F (u) ; )
f (u)
1 C(F (u) ; :::; F (u) ; )
(6)
is the hazard rate corresponding to the cumulative distribution function F C (u)
C(F (u) ; :::; F (u) ; ) on support [u (0) ; u (1)]. It is exactily as if the monopolist
is selling to consumers a choice of varieties. An appropriate regularity condition,
satis…ed for example in our FGM-exponential case, plays the same role as for singleproduct monopoly:
d (u + ) C (u; )
> 0:
du
A useful property of a copula family ordered by increasing orthant dependence is
that the conditional copula C1 (x; :::; x; ) increases (decreases) in when x is small
(large). This implies that greater positive dependence shifts up the hazard rate for
the multiproduct monopolist when market coverage is high enough
Lemma 1: Given increasing orthant dependence, there exists some u 2 (u(0); u (1)]
C
such that @ @(p; ) > 0 if p u .
Furthermore, it is straightforward that the market is fully covered, or nearly so,
if demand is su¢ ciently great.6 This consideration leads to the conclusion that
6
Let
for =
assumes
o
=
o
<
1
f (u(0))
o
and
.
o
u(0). Then the market is fully covered for
and almost fully covered
a small positive number. Our maintained interiority assumption implicitly
9
prices under multiproduct monopoly decrease with preference dependence if prefermm
ence strength is high. The pro…t of the multiproduct monopolist, mm
(pmm ),
however, always decreases with greater dependence, whether or not price increases,
because of the resulting downward shift in demand. Formally:
Proposition 1: Given increasing orthant dependence : (i) there exists
mm
mm
pmm > u (0) when =
and dpd < 0 if
; and (ii) d d < 0.
such that
Therefore, a multiproduct monopolist would prefer that consumer values for its
n products are less positively (more negatively) dependent. This is intuitive, since
the more similar are product varieties the less valuable is choice. Thus a higher
reduces quantity at any given price and hence reduces equilibrium pro…t, while the
e¤ect of on equilibrium price is more subtle. The lower quantity under a higher
motivates the …rm to lower price, but the slope of the demand curve also changes
with ; possibly having an opposing e¤ect on price. Both e¤ects work in the same
direction if demand is su¢ ciently strong. It is possible, however, that pmm increases
with if demand is su¢ ciently weak: For example, in the FGM-exponential case,
numerical analysis shows that pmm increases in if is below a critical value.
Now suppose that the n products are sold by n symmetric single-product oligopoly
…rms: Given that all other …rms charge price r; the pro…t function of Firm 1 is
n
(p; r) = (p + )Q (p; r:::; r) :
From (1),
@Q
=
@ p p=r
C1 (F (p) ; :::; F (p)) f (p) +
Z
1
(n
1) C12 (x; :::; x) f (u (x)) dx:
F (p)
In equilibrium, p = r = pn , satisfying
(pn + ) h(pn ; ) = 1;
10
(7)
where we de…ne the adjusted hazard rate for oligopoly competition
h(u; )
C
(u; ) + n (n
1)
R1
F (u)
1
C12 (x; :::; x; ) f (u (x)) dx
C (F (u) ; :::; F (u) ; )
;
(8)
which is equal to the hazard rate under multiproduct monopoly plus an extra term.
The extra term is the diversion ratio used in contemporary merger analysis (Shapiro
1996, Farrell and Shapiro, 2010), that is, the percentage demand increase from a price
cut resulting from customers who change allegiance. A modi…ed regularity condition,
once again satis…ed in the FGM-exponential case, guarantees a unique symmetric
equilibrium:
d (u + ) h(u; )
> 0:
(9)
du
Assuming the regularity condition for multiproduct monopoly holds, the regularity
condition for symmetric oligopoly additionally requires that the diversion ratio does
not fall too quickly as price rises. Each …rm’s equilibrium pro…t is
n
n
=
1
(pn + ) [1
n
C (F (pn ) ; :::; F (pn ) ; )] :
(10)
It is intuitive to expect that oligopoly competition intensi…es with more preference
dependence, as more consumers regard any two varieties to be close substitutes. In
general, however, the e¤ect of preference dependence on prices and pro…ts is ambiguous. As under multiproduct monopoly, the regularity condition is not enough
to ensure that prices monotonically decrease with . For while a higher shifts demand downward, motivating a lower price (market share e¤ect), it also may a¤ect the
slope of the residual demand curve, potentially providing an incentive to raise price
(price sensitivity e¤ect). Under oligopoly, a unilateral marginal reduction in price
impacts a …rm’s residual demand on both an extensive margin (market expansion)
and the intensive margin (business stealing). The ambiguity of the price sensitivity
e¤ect on the extensive margin explains why more substitutability between goods (e.g.
C12 (x; :::x; ) 0) may not be su¢ cient to conclude that pn decreases with .
We next identify su¢ cient conditions under which pn and n do decrease with .
)
The lemma below provides technical conditions that are su¢ cient for @h(p;
> 0:
@
11
Lemma 2 : Given increasing dependence, h(p; ) decreases in
h(u; ) +
and
d2 ln f (u)
du2
f 0 (u)
f (u)
if
0
nf 2 (u (x))
C11 (x; :::; x; ):
C (x; :::; x; )
(11)
(12)
Using the technical lemma, the next proposition identi…es su¢ cient conditions on
the copula and marginal distribution under which price and pro…ts under symmetric
n
n
< 0 and dd < 0. Part
oligopoly decrease in the degree of preference dependence: dp
d
(i) invokes positive stochastic dependence and limited log-curvature of the marginal
density (e.g. when f is approximately uniform or exponential). Part (ii) invokes
stronger log-curvature restrictions on the marginal density (e.g. when f is approximately uniform) without imposing restrictions on the copula.
Proposition 2 : If regularity condition (9) holds at pn , then, given increasing dependence, pn and n decrease in if either of the following conditions hold: (i) C11 < 0
2
2
f (u)
f (u)
both are not too negative.
and d ln
is su¢ ciently small; or (ii) d lnduf (u) and d ln
du2
du2
Propositions 1 and 2 suggest that preference dependence is a useful measure of
product di¤erentiation, disentangled from preference diversity. In fact, pro…ts actually
increase in preference diversity when is relatively small (Johnson and Myatt, 2006;
Chen and Riordan, 2011); whereas pro…ts always monotonically decrease in under
multiproduct monopoly, and pro…ts also monotonically decrease in under oligopoly
for all when f is approximately uniform or when f is approximately exponential
and C is positively dependent.
Thus the e¤ects of preference dependence ( ) on prices and pro…ts o¤er a new way
to think about product di¤erentiation. Both and can be interpreted as indicators of the degree of product di¤erentiation: higher indicates more heterogeneity of
consumer values for each product, while higher indicates greater similarity of these
values between products for a randomly chosen consumer. They have rather di¤erent economic meanings and the copula approach to modeling consumer preferences
disentangles their e¤ects in a general way. Proposition 2 loosely suggests that two
12
competing single-product …rms have a mutual incentive to coordinate the design or
promotion of their products so that consumer values are less positively dependent or
more negatively dependent.
IV. MARKET STRUCTURE AND PRICE
The copula approach also enables us to derive new results on how prices di¤er across
market structures, relating them to properties of the marginal distributions and the
dependence relationship. This provides new insights on how market structure a¤ects
…rm conduct. Speci…cally, we compare pm ; pn ; and pmm ; motivated by the scenarios
in Section 1.
We start with comparing the equilibrium oligopoly price with the single-product
monopoly price. While Chen and Riordan (2008) …nd su¢ cient conditions for pm T
pn when n = 2 and the marginal distribution is exponential (i.e. 0 ( ) = 0), it has
been an open question how the prices compare for arbitrary marginal distributions
and for any n 2. We can now answer with the following result:
Proposition 3 : If C11 < 0 (positive dependence) and 0 (p) 0; then pm > pn ; and if
C11 > 0 (negative dependence) and 0 (p) 0; then pm < pn :
Thus positive dependence and a non-decreasing hazard rate for the marginal distribution ensures that competition from other products lowers prices; conversely, negative
dependence and a non-increasing hazard rate ensures that oligopoly competition raises
price.7
This result can be understood as follows. An oligopolist sells less output at the
monopoly price, pm ; and thus a slight price reduction at pm is less costly to the
oligopolist since it applies to a smaller output. This "market share e¤ect" is a standard reason why one expects more competition to lower price. However, as Chen and
Riordan (2008) discuss in the context of a duopoly, there is a potentially o¤setting
"price sensitivity e¤ect" when products are di¤erentiated. Since an oligopolist sells
on a di¤erent margin from a monopolist, the slope of an oligopolist’s (residual) demand curve di¤ers from the slope of the single-product monopolist’s demand curve.
Furthermore, greater negative dependence makes it more di¢ cult for the oligopolist
7
Chen and Riordan (2007) and Perlo¤, Suslow, and Sequin (1995) present more speci…c models
of product di¤erentiation in which entry can result in higher prices.
13
to win over marginal consumers who value its own product less but its rival’s product
more. Similarly, a non-increasing hazard rate tends to put less consumer density
on the oligopolist’s intensive margin, further reducing price sensitivity.8 Together,
negative dependence and a non-increasing hazard rate are su¢ cient for the price sensitivity e¤ect to dominate the market share e¤ect, resulting in a higher price under
oligopoly competition.9
Although preference dependence and the number of …rms are di¤erent economic
concepts, our analysis suggests a common theme between their e¤ects on equilibrium
prices. Both greater preference dependence and more …rms represent increased competition. Each has a market share e¤ect— lower output— that favors lower prices,
but each may also have a price sensitivity e¤ect— potentially steepening the residual
demand curve— that favors higher prices. Propositions 2 and 3 give the respective
su¢ cient conditions for the net e¤ect to lower prices.
Next, we compare the prices for the multiproduct monopoly with those under
single-product monopoly and symmetric oligopoly.
Proposition 4 : If either C11 (x; :::; x) 0 for all x 2 (0; 1) or C11 (x; :::; x)
x 2 (0; 1), then pmm > pn and pmm > pm :
0 for all
As one might expect, pmm > pn ; or prices for n substitutes are higher under
monopoly than under competition, extending the result for n = 2 in Chen and Riordan (2008). The familiar intuition is that a multiproduct monopolist internalizes the
negative e¤ects of reducing one product’s price on pro…ts from the other products.
The comparison of prices under multiproduct monopoly (pmm ) and single-product
monopoly (pm ) is more subtle. The multiproduct monopolist has higher total output
at pm than the single-product monopolist, which motivates it to raise its symmetric price above pm : But, as with the oligopoly comparison, the marginal consumers
of the multiproduct monopolist di¤er from those of the single-product monopolist,
8
When n = 2; the argument in the proof of Proposition ?? can be adapted to show more formally
that, with 0 ( )
0; the (residual) demand curve of a duopolist is indeed steeper than that of
the monopolist if C ( ; ) is negatively dependent, independent, or has su¢ ciently limited positive
dependence.
9
Note that due to more varieties under oligopoly, a higher price under oligopoly does not imply
that consumer welfare is lower under oligopoly competition than under single-product monopoly.
14
which can potentially make the slope of the multiproduct monopolist’s demand curve
steeper than that of the single-product monopolist. Interestingly, the market share
e¤ect unambiguously dominates, provided that C (x; :::x) exhibits uniform positive
or negative stochastic dependence.
Under general preference distributions, Propositions 3 and 4 largely settle the issue
of how prices in symmetric multiproduct industries compare to the single-product
monopoly price.
V. CONCLUSION
Using copulas to describe the distribution of consumer preferences is a convenient
and intuitive approach to discrete choice demand. The approach enables us to
identify preference dependence as a distinct indicator of product di¤erentiation in
multiproduct industries, disentangled from the e¤ects of preference diversity, in the
sense that greater correlation of consumer values for alternative products leads to
lower prices and pro…ts under certain conditions. The approach also leads to new
results in price theory. The entry of symmetric di¤erentiated competitors into an
initial single-product monopoly lowers (raises) price if preferences are positively (negatively) dependent and the hazard rate of the marginal distribution is non-decreasing
(non-increasing). Moreover, under a uniform dependence condition, price rises when
a single-product monopolist adds symmetric di¤erentiated varieties to its product
line.
There are several directions for further research for which the copula approach is
likely to be useful. One is to examine further the e¤ects of entry into di¤erentiated
product markets. Whereas we have found that entry into an intitial monopoly raises
or lowers price depending on preference dependence and the hazard rate, it is important additionally to determine the conditions under which the symmetric oligopoly
price is decreasing or increasing in the number of …rms. Similarly, it is important to
understand the conditions under which a product line expansion by a multiproduct
monopolist results in higher or lower prices. Also, relaxing symmetry is important,
even though this is likely to challenge tractability. For example, a symmetric model
seems inappropriate for understanding conditions under which generic entry results
in higher or lower branded drug prices (Perlo¤, Suslow, and Seguin, 1995).
15
The copula representation of consumer preferences may be valuable for studying
other applied microeconomics topics. Chen and Riordan (2013) applies the copula
approach to study the pro…tability of product bundling, and further applications
might shed more light on the positive and normative economics of bundling. Chen
and Pearcy (2010) uses a speci…c class of copulas to model intertemporal dependence
of consumer values. Other promising topic areas include the economics of search
(e.g., Anderson and Renault 1999; Schultz and Stahl 1996; Bar Isaac, Caruana, and
Cunat 2010), and the endogenous determination of market structure (e.g., Shaked and
Sutton, 1990). Furthermore, the copula approach to discrete choice demand, and its
potentially rich set of predictions about market structure, conduct, and performance,
might open interesting new directions for empirical industrial organization research.10
REFERENCES
Anderson, S.P.; de Palma, A. and Thisse, J-F., 1992, Discrete Choice Theory of
Product Di¤erentiation (MIT Press, Cambridge, Massachusetts, U.S.A.).
Anderson, S.P. and Renault, R., 1999, ‘Pricing, Product Diversity, and Search Costs:
A Bertrand-Chamberlin-Diamond Model,’RAND Journal of Economics, 30, pp, 71935.
Bar-Isaac, H.; Caruana, G and Cuñat, V., 2012, ‘Search, Design, and Market Structure,’American Economic Review, 102, pp. 1140-60.
Chen, Y. and Pearcy, J., ‘Dynamic pricing: when to entice brand switching and when
to reward consumer loyalty,’RAND Journal of Economics, 41, pp. 674-685.
Chen, Y. and Riordan, M.H., 2008, ‘Price-increasing Competition,’ RAND Journal
of Economics, 39, pp. 1042-1058.
__________, 2007, ‘Price and Variety in the Spokes Model,’Economic Journal,
117, pp. 897-921.
__________, 2011, ‘Preferences, Prices, and Performance in Multiproduct Industries,’Columbia University working paper.
10
See, for example, Chen and Savage (2011) for an empirical analysis of how preference dispersion
a¤ects prices and price di¤erences between monopoly and duopoly markets for Internet services.
16
__________, 2013, ‘Pro…tability of Product Bundling,’International Economic
Review, 54, pp. 35-57.
Chen, Y. and Savage, S., 2011, ‘The E¤ects of Competition on the Price for Cable
Modem Internet Access,’Review of Economics and Statistics, 93, pp. 201-217.
Farrell J. and Shapiro, C., 2010, ‘Antitrust Evaluation of Horizontal Mergers: An Economic Alternative to Market De…nition,’The B.E. Journal of Theoretical Economics,
Policy and Perspectives, 10, pp. 1-41.
Joe, H., 1997, Multivariate Models and Dependence Concepts ( Chapman and Hall,
London, U.K.).
Johnson, J.P. and Myatt, D.P., 2006, ‘On the Simple Economics of Advertising,
Marketing, and Product Design,’American Economic Review, 96, pp. 756-784.
Nelsen, R.B., 2006, An Introduction to Copulas (Springer, New York, U.S.A.).
Perlo¤, J.M. and Salop, S.C., 1985, ‘Equilibrium with Product Di¤erentiation,’Review of Economic Studies, LII, pp. 107-120.
Perlo¤, J.M.; Suslow, V.Y. and Seguin, P.M., 1995, ‘Higher prices from Entry: Pricing
of Brand-name Drugs,’W.P. No. CPC-99-03, Competition Policy Center, University
of California at Berkeley.
Shaked, A. and Sutton, J., 1990, ‘Multiproduct Firms and Market Structure,’RAND
Journal of Economics, 21, pp. 45-62.
Shapiro, C., 1996, ‘Mergers with Di¤erentiated Products,’Antitrust, Spring, pp. 2330.
APPENDIX
The appendix contains proofs for Lemma 1, Proposition 1, Lemma 2, Propositions
2, 3, and 4.
Proof of Lemma 1. Given increasing orthant dependence,
C (F (p); :::; F (p); ) = n
Z
0
17
F (p)
C1 (x; :::; x; ) dx
increases in for any p > F 1 (0) ; which is possible only if C1 (x; :::; x; ) > 0 for all
when x is close to zero. Similarly, C1 (x; :::; x; ) < 0 for all if x is su¢ ciently
close to 1. Thus there must exist x0 > 0 such that, for all ; C1 (x0 ; :::; x0 ; ) = 0 and
C1 (x; :::; x; ) > 0 if x < x0 .11 Since
@
C
(p; )
@
C1 (F (p) ; :::; F (p) ; )
C1 (F (p) ; :::; F (p) ; ) C (F (p) ; :::; F (p) ; )
= n
+
f (p) ;
1 C (F (p) ; :::; F (p) ; )
[1 C (F (p) ; :::; F (p) ; )]2
there exists some u 2 [F
1
(x0 ) ; u (1)] such that @
C
(p; ) =@ > 0 for all
if p
u
Proof of Proposition 1. (i) From (4), for any ; let be such that [u + ] C (u ; ) =
1, where u
F 1 (x01 ) > u (0) is de…ned in Lemma 1. Then, pmm = u > u (0) if
C mm
= . If
, then pmm u and Lemma 1 implies @ (p@ ; ) > 0. It follows
mm
mm
from (4) and (5) that dpd < 0 and hence dpd < 0: (ii) holds from application of
the envelope theorem to (3) and C > 0:
Proof of Lemma 2. Notice that (suppressing the argument to simplify notation),
dC1 (x; :::; x)
= C11 (x; :::; x) + (n
dx
or
(n
1) C12 (x; :::; x) =
dC1 (x; :::; x)
dx
Thus,
h (p) =
C
(p) + n (n
1)
R1
R1
F (p)
h
1
1) C12 (x; :::; x) ;
C11 (x; :::; x) :
C12 (x; :::; x) f (u (x)) dx
C (F (p) ; :::F (p))
dC1 (x;:::;x)
dx
i
C11 (x; :::; x) f (u (x)) dx
F (p)
nC1 (F (p) ; ::::F (p)) f (p)
+n
1 C (F (p) ; :::; F (p))
1 C (F (p) ; :::; F (p))
R 1 dC1 (x;:::;x)
R1
f (u (x)) dx
C (x; :::; x)f (u (x)) dx
nC1 (F (p) ; :::; F (p)) f (p)
dx
F (p)
F (p) 11
=
+n
:
1 C (F (p) ; :::; F (p))
1 C (F (p) ; :::; F (p))
=
11
Similarly, there exists x00
n = 2, x00 = x0 = 1=2.
x0 such that C1 (x; :::; x; ) < 0 if x > x0 : For the FGM family with
18
Since
Z
1
F (p)
dC1 (x; :::; x)
f (u (x)) dx
dx
= f (u (1))
Z
C1 (F (p) ; :::; F (p)) f (p)
1
C1 (x; :::; x)
F (p)
f 0 (u (x))
dx;
f (u (x))
we have
h (p) = n
= n
=
R1
f (u (1))
F (p)
f (u (1)) +
f (u (1))
f (p)
+n
f (p)
1
R1
1
n
0
0
(u(x))
C1 (x; :::; x) ff (u(x))
dx
R1
[1
F (p)
C11 (x; :::; x)f (u (x)) dx
F (p)
C (F (p) ; :::F (p))
R1
C (x; :::; x)f (u (x)) dx
F (p) 11
f 0 (u(x)) 1 d[1 C(x;:::;x)]
dx
F (p) f (u(x)) n
1
R1
C (F (p) ; :::; F (p))
C (x; :::; x)]
1
d
f 0 (u(x))
f (u(x))
dx
R1
F (p)
C11 (x; :::; x)f (u (x)) dx
C (F (p) ; :::; F (p))
:
Therefore, if (11) holds, then
Z1 h
C (x;:::x) d2 ln f (u)
f (u(x))
du2
nC11
@h (p) F (p)
=
@
i
h
(x; :::; x)f (u (x)) dx + h (p) +
1
f 0 (p)
f (p)
i
C (F (p) ; :::F (p))
>0
C (F (p) ; :::; F (p))
because C > 0 and C11 < 0 by increasing orthant dependence and increasing stochastic dependence respectively.
n
n
Proof of Proposition 2. First, observe that dp
has the same sign as dp
. Second,
d
d
dpn
n
observe that
decreases in when d < 0 because
d
d
n
=
=
@
@
n
+
@ n dpn
@ pn d
1 n
(p + ) C
n
F pd ; :::; F pd ;
+
@ n dpn
< 0;
@ pn d
n
where C F pd ; :::; F pd ; > 0 from increasing orthant dependence; and @@ pn > 0
by the envelope theorem and by the fact that a …rm’s demand increases in the other
d
…rm’s price. Given these observations we focus on su¢ cient conditions for dp
< 0.
d
19
)
Since @h(p;
> 0 and the regularity condition imply
@
the conditions for Lemma 2.
2
f (u)
! 0 and C11 (x:::x; ) < 0, then
(i) If d ln
du2
Z1
f (u (1))
h (p) +
2
0
f (p)
!n
f (p)
dp
d
< 0, it is su¢ cient to verify
C11 (x; :::; x; )f (u (x)) dx
F (p)
1
C (F (p) ; :::; F (p) ; )
>0
2
f (u)
(u(x))
C (x; :::; x; ) since C11 (x; :::x; ) < 0; thus satisfying Lemma
and d ln
> 2f
du2
C (x;:::x) 11
2:
2
0 (u)
f (u)
(ii) If d lnduf (u) and d ln
both are not too negative, then h (u) + ff (u)
0 and
du2
d2 ln f (u)
du2
2
(u(x))
> C2f (x;:::;x)
C11 (x; :::; x) by increasing stochastic dependence, thus satisfying
Lemma 2.
Proof of Proposition 3. It su¢ ces to show that (i) h (p) > (p) if C11 < 0 and
0
(p) 0; and (ii) h (p) < (p) if C11 > 0 and 0 (p) 0:
(i) Suppose that 0 (p) 0: Then, since
dC1 (x1 ; :::; x1 )
dx1
C11 (x1 ; :::; x1 ) = (n
nC1 (F (p) ; :::; F (p)) f (p)
+n
h (p) =
1 C (F (p) ; :::; F (p))
nC1 (F (p) ; :::; F (p)) f (p)
=
+n
1 C (F (p) ; :::; F (p))
R1
F (p)
(1
R1
(1
F (p)
nC1 (F (p) ; :::; F (p)) f (p)
f (p)
+n
1 C (F (p) ; :::; F (p))
1 F (p)
1) C12 (x1 ; :::; x1 ) > 0;
1) C12 (x; :::; x) 1 f F(u(x))
dx
(u(x))
x) (n
1
x)
h
C (F (p) ; :::; F (p))
dC1 (x;:::;x)
dx
i
f (u(x))
dx
1 F (u(x))
C (F (p) ; :::; F (p))
h
i
dC1 (x;:::;x)
(1
x)
C
(x;
:::;
x)
dx
11
dx1
F (p)
R1
1
C11 (x; :::; x)
1
C (F (p) ; :::; F (p))
Substituting
Z
1
F (p)
= j(1
dC1 (x; :::; x)
dx
Z
1
x) C1 (x; :::; x) F (p) +
(1
x)
C11 (x; :::; x) dx
1
C1 (x; :::; x) dx
F (p)
Z
1
F (p)
20
(1
x) C11 (x; :::; x) dx
:
=
(1
1
F (p)) C1 (F (p) ; :::; F (p))+ (1
n
C (F (p) ; :::; F (p)))
Z
1
(1
x) C11 (x; :::; x) dx;
F (p)
and simplifying, we obtain
"
h (p)
(p) 1
f (p)
n
1 F (p)
R1
F (p)
1
(1
x) C11 (x; :::; x) dx
C (F (p) ; :::; F (p))
#
:
#
:
Hence h (p) > (p) if in addition C11 < 0:
(ii) Suppose that 0 0: By analogous derivations, we have
h (p) <
"
(p) 1
f (p)
n
1 F (p)
R1
F (p)
1
(1
x) C11 (x; :::; x) dx
C (F (p) ; :::; F (p))
Hence h (p) < (p) if in addition C11 (x; :::; x) > 0:
Proof of Proposition 4. (i) Since h (p) > C (p) from (8); comparing (4) and (7)
leads to pn < pmm .
(ii) It su¢ ces to show that C (p) < (p) for all p 2 (u (0) ; u (1)) if C11 (x; :::; x)
0; C11 (x; :::; x) = 0; or C11 (x; :::x) 0 for all x 2 (0; 1) :
First,
C
(p)
=
(p)
If C11
Z
nC1 (F (p);:::;F (p))
f
1 C(F (p);:::;F (p))
f (p)
1 F (p)
F (p)
=
nC1 (F (p) ; :::; F (p))
[1
1 C(F (p) ; :::; F (p))
F (p)] :
0 or if C11 = 0; then, since
1
(1
(p)
x) dC1 (x; :::; x) =
(1
1
F (p)) C1 (F (p) ; :::; F (p))+ [1
n
21
C (F (p) ; :::; F (p))]
and C1k (x; :::; x) = C12 (x; :::; x) > 0 for x 2 (0; 1) and for all k 6= 1;
[1
(p)
=
(p)
C (F (p) ; :::; F (p))]
C
=
1
R1
F (p)
(1
x) dC1 (x; :::; x)
C(F (p) ; :::; F (p))
R1
C (F (p) ; :::; F (p))] n F (p) (1 x) [C11 (x; :::; x) + (n
[1
= 1
n
n
1
R1
F (p)
(1
C(F (p) ; :::; F (p))
x) [C11 (x; :::; x) + (n
1
1)C12 (x; :::; x)] dx
1)C12 (x; :::; x)] dx
C(F (p) ; :::; F (p))
< 1:
Next, suppose C11 (x; :::; x) 0 for all x 2 (0; 1) ; so that there is positive stochastic
dependence: Then, C1 (x; :::; x) C(x;:::;x)
for all x 2 (0; 1) since
x
C (x; :::; x) =
Zx
C1 (t; x; :::; x) dt
0
Zx
C1 (x; :::; x) dt = C1 (x; :::; x) x:
0
Hence, letting x = F (p) ;
C
(u (x))
n (1
=
(u (x))
1
x) C1 (x; :::; x)
C (x; :::; x)
n (1 x) C (x; :::; x)
:
x [1 C (x; :::; x)]
C
(u(x))
1 (x;:::;x)
= n(11 x)C
1: Then
Now, suppose to the contrary that (u(x))
C(x;:::;x)
1: We will show that this leads to a contradiction. First, notice that
lim
x!1
n (1 x) C (x; :::; x)
= n lim
x!1
x [1 C (x; :::; x)]
n(1 x)C(x;:::;x)
x[1 C(x;:::;x)]
C (x; :::; x) + (1 x) nC1 (x; :::; x)
= 1:
1 C (x; :::; x) nxC1 (x; :::; x)
Next, letting x = (x; :::x) to simplify notation; we have
d
=
=
h
(1 x)C(x;:::;x)
x[1 C(x;:::;x)]
i
dx
[ C (x) + n (1
n (1
x) C1 (x)] fx [1
x) C1 (x) x [1
C (x)]g (1 x) C (x) [1 C (x) nxC1 (x)]
fx [1 C (x)]g2
C (x)] C (x) [1 C (x)] + (1 x) C (x) nxC1 (x)
fx [1 C (x)]g2
22
x) C1 (x) x C (x) [1 C (x)]
[1
2
fx [1 C (x)]g
[1 C (x)] [x C (x)]
> 0 for x 2 (0; 1) ;
=
fx [1 C (x)]g2
=
n (1
C (x)] x
fx [1
C (x) [1 C (x)]
C (x)]g2
1 (x;:::;x)
where the …rst inequality is due to n(11 x)C
1 by assumption, and the second
C(x;:::;x)
inequality holds because x > C (x) : It follows that, for any interior x;
n [1 x] C (x; :::x)
< 1;
x [1 C (x; :::x)]
which is a contradiction. Therefore
for any p 2 (u (0) ; u (1)) :
n[1 x]C1 (x;:::;x)
1 C(x;:::;x)
23
< 1 for any x 2 (0; 1) ; or
C
(p)
(p)
<1